Notes

Any pattern grown from a single cell according to rules that do not distinguish different directions on a lattice must show the same symmetry as the lattice. But we have seen that in fact many rules actually yield almost circular patterns with much higher symmetry. One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n:

Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]]

For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to

For odd n this is inevitably true for any lattice with mirror symmetry. But for even n it can fail. For a square lattice, it still nevertheless always holds up to n=2 (so that the analogs of moments of inertia satisfy Ιxx == Ιyy, Ιxy == Ιyx == 0). And for a hexagonal lattice it holds up to n=4. But when n=4 isotropy requires the {1,1,1,1} and {1,1,2,2} tensor components to have ratio β=3—while square symmetry allows these components to have any ratio. (In general there will be more than one component unless the representation of the lattice symmetry group carried by the rank n tensor is irreducible.) In 3D no regular lattice forces isotropy beyond n=2, while in 4D the SO(8) lattice works up to n=4, in 8D the E8 lattice up to n=6, and in 24D the Leech lattice up to n=10. (Lattices that give dense sphere packings tend to show more isotropy.) Note that isotropy can also be characterized using analogs of multipole moments, obtained in 2D by summing Subscript[r,i] Exp[ⅈ n Subscript[θ,i]], and in higher dimensions by summing appropriate SphericalHarmonicY or GegenbauerC functions. For isotropy, only the n=0 moment can be nonzero. On a 2D lattice with m directions, all moments are forced to be zero except when m divides n. (Sums of squares of moments of given order in general provide rotationally invariant measures of anisotropy—equal to pair correlations weighted with LegendreP or GegenbauerC functions.)

Even though it is not inevitable from lattice symmetry, one might think that if there is some kind of effective randomness in the underlying rules then sufficiently large patterns would still often show some sort of average isotropy. And at least in the case of ordinary random walks, they do, so that for example, the ratio averaged over all possible walks of n=4 tensor components after t steps on a square lattice is β=3 + 2/(t-1), converging to the isotropic value 3, and the ratio of n=6 components is 5 - 4/(t-1) + 32/(3t-4). For the aggregation model of page 331, β also decreases with t, reaching 4 around t=10, but now its asymptotic value is around 3.07.

In continuous systems such as partial differential equations, isotropy requires that coordinates in effect appear only in ∇. In most finite difference approximations, there is presumably isotropy in the end, but the rates of convergence are almost inevitably rather different in different directions relative to the lattice.